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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Non-vanishing of derivatives of $ GL(3) \times GL(2)$ $ L$-functions


Author: Qingfeng Sun
Journal: Proc. Amer. Math. Soc. 140 (2012), 449-463
MSC (2010): Primary 11F67, 11F12, 11F30
Published electronically: June 14, 2011
MathSciNet review: 2846314
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Abstract: Let $ f$ be a fixed self-dual Hecke-Maass cusp form for $ SL_3(\mathbb{Z})$ and let $ \mathcal{B}_k$ be an orthogonal basis of holomorphic cusp forms of weight $ k \equiv 2(\mathrm{mod} 4)$ for $ SL_2(\mathbb{Z})$. We prove an asymptotic formula for the first moment of the first derivative of $ L\left(s,f\times g\right)$ at the central point $ s=1/2$, where $ g$ runs over $ \mathcal{B}_k$, $ K\leq k\leq 2K$, $ K$ large enough. This implies that for each $ K$ large enough there exists $ g\in \mathcal{B}_k$ with $ K\leq k\leq 2K$ such that $ L'(1/2,f\times g)\neq 0$.


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Additional Information

Qingfeng Sun
Affiliation: School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, People’s Republic of China
Email: qfsun@mail.sdu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10947-X
PII: S 0002-9939(2011)10947-X
Keywords: Non-vanishing, $GL(3) ×GL(2)$ $L$-functions
Received by editor(s): April 2, 2010
Received by editor(s) in revised form: September 24, 2010, November 28, 2010, and November 29, 2010
Published electronically: June 14, 2011
Additional Notes: The author was supported by National Natural Science Foundation of China (grant No. 10971119).
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.